Research on the Dynamic Responses of Simply Supported Horizontal Pipes Conveying Gas-Liquid Two-Phase Slug Flow

: The dynamic responses of simply supported horizontal pipes conveying gas-liquid two-phase slug ﬂow are explored. The intermittent characteristics of slug ﬂow parameters are mainly considered to analyze the dynamic model of the piping system. The results show that the variations of the midpoint transverse displacement could vary from periodic-like motion to a kind of motion whose amplitude increases as time goes on if increasing the superﬁcial gas velocity. Meanwhile, the dynamic responses have certain relations with the vibration acceleration. By analyzing the parameters in the power spectrum densities of vibration acceleration such as the number of predominant frequencies and the amplitude of each peak frequency, the dynamic behaviors of the piping system like periodicity could be calculated expediently. data


Introduction
Fluid-conveying piping systems have been continually adopted in various processes of modern chemical engineering. It is crucial to guarantee the safety of the process of transporting materials. As is known, instability of the piping system may occur due to the vibrations induced by the internal flow, which has drawn the attention of numerous scholars. The explorations of the dynamics of the fluid-conveying piping system since the 1880s have been summarized systematically by Paidoussis and Issid [1].
It has been noted that with a sufficiently large constant flow velocity, divergence or flutter could happen for the piping system. In addition, for harmonically varying flow velocity, some complex dynamical behaviors like quasi-periodic may take place, which would bring more damages to the operation of the pipeline, which have been further confirmed by the researches of Ariaratnam and Namachchivaya [2] and Jin and Song [3]. Pipes conveying single-phase fluid have been researched deeply by scholars including the above several experts.
Actually, piping systems conveying two-phase flow may have a wider range of applications in science and engineering [4]. Alamian et al. [5] established a mathematical model to explore the instantaneous flow inside the gas pipelines. They discussed the effects of boundary conditions particularly. Goodarzi et al. [6] and [7] investigated the Erosion phenomenon in pipes conveying two-phase flow. Pourfattah et al. [8] carried out the twophase flow simulation of the heat transfer characteristics of the manifold microchannel heat sink. Almasi et al. [9] introduced an SPH method to explore the multiphase phenomenon. Shadloo et al. [10] discussed the pressure drop which is one significant parameter that As shown in Figure 2, a gas bubble and a liquid slug would constitute one stable slug unit. It is assumed that the sectional area of the inner pipe is Ai and the volume flow rate is Q. Then the superficial gas velocity uSG and superficial liquid velocity uSL are obtained by: Referring to the work of Wang [25], the mathematical model of a slug flow conveying pipe could be written as: The implications of the parameters of the above model could be found in the paper of Liu and Wang [26].
The transient pattern of slug flow at two typical moments is shown in Figure 3. It could be noted that uL(x,t) = uLF at t = t1 as shown in Figure 3a while uL(x,t) = uS at t = t2 as depicted in Figure 3b, which means that uL(x,t) is the function of the coordinate x and the time t. Then it is pivotal to obtain the variations of flow parameters with x and t. to analyze Equation (2), which could be found in Liu and Wang [26]. As shown in Figure 2, a gas bubble and a liquid slug would constitute one stable slug unit. It is assumed that the sectional area of the inner pipe is Ai and the volume flow rate is Q. Then the superficial gas velocity u SG and superficial liquid velocity u SL are obtained by:  As shown in Figure 2, a gas bubble and a liquid slug would constitute one stable slug unit. It is assumed that the sectional area of the inner pipe is Ai and the volume flow rate is Q. Then the superficial gas velocity uSG and superficial liquid velocity uSL are obtained by: Referring to the work of Wang [25], the mathematical model of a slug flow conveying pipe could be written as: The implications of the parameters of the above model could be found in the paper of Liu and Wang [26].
The transient pattern of slug flow at two typical moments is shown in Figure 3. It could be noted that uL(x,t) = uLF at t = t1 as shown in Figure 3a while uL(x,t) = uS at t = t2 as depicted in Figure 3b, which means that uL(x,t) is the function of the coordinate x and the time t. Then it is pivotal to obtain the variations of flow parameters with x and t. to analyze Equation (2), which could be found in Liu and Wang [26]. Referring to the work of Wang [25], the mathematical model of a slug flow conveying pipe could be written as: The implications of the parameters of the above model could be found in the paper of Liu and Wang [26].
The transient pattern of slug flow at two typical moments is shown in Figure 3. It could be noted that u L (x,t) = u LF at t = t 1 as shown in Figure 3a while u L (x,t) = u S at t = t 2 as depicted in Figure 3b, which means that u L (x,t) is the function of the coordinate x and the time t. Then it is pivotal to obtain the variations of flow parameters with x and t. to analyze Equation (2), which could be found in Liu and Wang [26]. As shown in Figure 2, a gas bubble and a liquid slug would constitute one stable slug unit. It is assumed that the sectional area of the inner pipe is Ai and the volume flow rate is Q. Then the superficial gas velocity uSG and superficial liquid velocity uSL are obtained by: Referring to the work of Wang [25], the mathematical model of a slug flow conveying pipe could be written as: The implications of the parameters of the above model could be found in the paper of Liu and Wang [26].
The transient pattern of slug flow at two typical moments is shown in Figure 3. It could be noted that uL(x,t) = uLF at t = t1 as shown in Figure 3a while uL(x,t) = uS at t = t2 as depicted in Figure 3b, which means that uL(x,t) is the function of the coordinate x and the time t. Then it is pivotal to obtain the variations of flow parameters with x and t. to analyze Equation (2), which could be found in Liu and Wang [26].  The liquid flow velocity u L (0.5 L,t) is depicted in Figure 4, which could reflect the features of intermittency. This characteristic will lead to some flow parameters in Equation (2) varying with time, which would affect the dynamic responses of the pipes greatly.  The liquid flow velocity uL(0.5 L,t) is depicted in Figure 4, which could reflect the features of intermittency. This characteristic will lead to some flow parameters in Equation (2) varying with time, which would affect the dynamic responses of the pipes greatly. Galerkin's method is employed in this work with the transverse displacement being written as: where φi(x) are the eigenfunctions of the simply supported beam and qi(t) are the generalized coordinates. Two-mode expansion (n = 2) is utilized in this study.
By means of substituting Equation (3) into Equation (2), left multiplying by and integrating it from 0 to L, the following matrix equation can be obtained after a series of operations: where M, C and K are all two-order matrixes. The expressions of these matrixes could be referred to Liu and Wang [26]. It is supposed that = [ ,̇] . Then Equation (4) is transferred to a differential equation： ̇= The coefficient matrix E is written as where I is the two-order unit matrixes. Considering the nonlinear factor, Equation (5) is a differential equation set that is calculated by the Runge Kutta method in this work. Then the variations of vibration parameters including vibration displacement, velocity, and acceleration with time could be obtained.

Results and Discussion
In this study, the inner diameter of the pipe Di = 0.025 m, the outer diameter of the pipe D = 0.028 m, the length of the pipe is 10 m, the density of the fluid ρL = 998 kg/m 3 , and the density of the gas ρG = 1.20 kg/m 3 . uSL would be fixed as 1.0 m/s with uSG varying from Galerkin's method is employed in this work with the transverse displacement being written as: where ϕ i (x) are the eigenfunctions of the simply supported beam and q i (t) are the generalized coordinates. Two-mode expansion (n = 2) is utilized in this study. By means of substituting Equation ( q+Kq=0 (4) where M, C and K are all two-order matrixes. The expressions of these matrixes could be referred to Liu and Wang [26]. It is supposed that p = q, . q T . Then Equation (4) is transferred to a differential equation: The coefficient matrix E is written as where I is the two-order unit matrixes. Considering the nonlinear factor, Equation (5) is a differential equation set that is calculated by the Runge Kutta method in this work. Then the variations of vibration parameters including vibration displacement, velocity, and acceleration with time could be obtained.

Results and Discussion
In this study, the inner diameter of the pipe Di = 0.025 m, the outer diameter of the pipe D = 0.028 m, the length of the pipe is 10 m, the density of the fluid ρ L = 998 kg/m 3 , and the density of the gas ρ G = 1.20 kg/m 3 . u SL would be fixed as 1.0 m/s with u SG varying from 3.0 m/s to 10.0 m/s. The flow regimes of these conditions are all slug flow based on the flow pattern map mentioned in Zhang et al. [27]. In the following discussion, two kinds of pipe including Young's modulus E = 70 GPa and E = 120 GPa will be analyzed successively. The variations of the midpoint transverse displacement will be employed to represent the dynamic responses of the piping system.

The Dynamic Responses of E = 70 GPa
The bifurcation diagram of E = 70 GPa is constructed to explore the dynamical behaviors of the piping system at different u SG and u SL , which can be obtained by recording the midpoint displacement when the midpoint velocity is zero as shown in Figure 5. It is observed that the dynamical behaviors are similar to each other when u SG is less than 7.0 m/s. However, a great change of the dynamical behaviors would happen when u SG is larger than 7.0 m/s. Due to this, we will discuss the dynamic responses of  [27]. In the following discussion, two kinds of pipe including Young's modulus E = 70 GPa and E = 120 GPa will be analyzed successively. The variations of the midpoint transverse displacement will be employed to represent the dynamic responses of the piping system.

The Dynamic Responses of E = 70 GPa
The bifurcation diagram of E = 70 GPa is constructed to explore the dynamical behaviors of the piping system at different uSG and uSL, which can be obtained by recording the midpoint displacement when the midpoint velocity is zero as shown in Figure 5. It is observed that the dynamical behaviors are similar to each other when uSG is less than 7.0 m/s. However, a great change of the dynamical behaviors would happen when uSG is larger than 7.0 m/s. Due to this, we will discuss the dynamic responses of the piping system under six conditions where uSG are 3.0, 5.0, 7.0, 8.0 9.0, 10.0 m/s respectively. The dynamic responses of the piping system for the case of E = 70 GPa and uSL = 1.0 m/s at different uSG are shown in Figure 6. It is observed from Figure 6a,b that the variations of the midpoint transverse displacement at uSG = 3.0 m/s and uSG = 5.0 m/s are both similar to a kind of cyclic motion but it is not the normal cyclic motion like simple harmonic motion on account of the amplitude of the piping system varying with time. The variation of the amplitude value at certain moments for uSG = 5.0 m/s is greater than that of uSG = 3.0 m/s. Nevertheless, some differences may exist for the variations of the period of these two cases. With the increase of uSG, the variations of the midpoint transverse displacement would change considerably. For the case of uSG = 7.0 m/s as shown in Figure 6c, the absolute value of the amplitude increases gradually with the lapse of time in the first 30 seconds. However, the decrease of the amplitude would take place in the second 30 seconds. Although we mainly discuss the dynamic responses of the first 60 seconds in this study, it can be speculated that the increase and decrease may still continue to appear alternately as time goes on further. It is worth noting that the variations of the midpoint transverse displacement of uSG = 8.0 m/s are different from those of uSG = 7.0 m/s completely as depicted in Figure 6d. The increase and decrease of the absolute value of the amplitude appear alternately within a short time. When uSG increases to 9 m/s as shown in Figure 6e, the midpoint transverse displacement seems to perform a similar varying pattern of the case of uSG = 7.0 m/s. At the beginning stage of the vibration, the absolute value of the amplitude increases first and then decreases just like the condition of uSG = 7.0 m/s. This state will continue about four times for 40 seconds. Yet the amplitude will continue to increase as time goes on. The condition of uSG = 10.0 m/s as shown in Figure 6f is similar The dynamic responses of the piping system for the case of E = 70 GPa and u SL = 1.0 m/s at different u SG are shown in Figure 6. It is observed from Figure 6a,b that the variations of the midpoint transverse displacement at u SG = 3.0 m/s and u SG = 5.0 m/s are both similar to a kind of cyclic motion but it is not the normal cyclic motion like simple harmonic motion on account of the amplitude of the piping system varying with time. The variation of the amplitude value at certain moments for u SG = 5.0 m/s is greater than that of u SG = 3.0 m/s. Nevertheless, some differences may exist for the variations of the period of these two cases. With the increase of u SG , the variations of the midpoint transverse displacement would change considerably. For the case of u SG = 7.0 m/s as shown in Figure 6c, the absolute value of the amplitude increases gradually with the lapse of time in the first 30 seconds. However, the decrease of the amplitude would take place in the second 30 seconds. Although we mainly discuss the dynamic responses of the first 60 seconds in this study, it can be speculated that the increase and decrease may still continue to appear alternately as time goes on further. It is worth noting that the variations of the midpoint transverse displacement of u SG = 8.0 m/s are different from those of u SG = 7.0 m/s completely as depicted in Figure 6d. The increase and decrease of the absolute value of the amplitude appear alternately within a short time. When u SG increases to 9 m/s as shown in Figure 6e, the midpoint transverse displacement seems to perform a similar varying pattern of the case of u SG = 7.0 m/s. At the beginning stage of the vibration, the absolute value of the amplitude increases first and then decreases just like the condition of u SG = 7.0 m/s. This state will continue about four times for 40 seconds. Yet the amplitude will continue to increase as time goes on. The condition of u SG = 10.0 m/s as shown in Figure 6f is similar to the condition of u SG = 9.0 m/s. In summary, with the increase of superficial gas velocity, the dynamic responses will vary from periodic-like motion to a kind of motion whose amplitude increases with time.
to the condition of uSG = 9.0 m/s. In summary, with the increase of superficial gas velocity, the dynamic responses will vary from periodic-like motion to a kind of motion whose amplitude increases with time.  It can be immediately found that visible distinctions exist with an increase of uSG. In Figure 7a, three predominant frequencies could be observed with the amplitude decreasing as the peak frequency increases. When the superficial gas velocity is 5.0 m/s, four predominant frequencies exist as shown in Figure 7b, which could prove that some differences may exist for the variations of the period of these two cases as mentioned above. The amplitudes of the minimum peak frequency and the maximal peak frequency are almost the same. While the amplitudes of second and third peak frequencies are relatively small. The condition of uSL = 7.0 m/s as depicted in Figure 7c seems to be similar to the condition of uSG = 5.0 m/s. However, the amplitude of the maximal peak frequency is larger than that of the minimum peak frequency. It was noted that the midpoint transverse displacement would no longer present the characteristic of periodicity when the superficial gas velocity is larger than 7.0 m/s. Some interesting phenomena will also occur for these conditions. Firstly, four predominant frequencies exist in the case of uSG = 8.0 m/s as shown in Figure 7d. The amplitudes of first and second peak frequencies are both very small. As the peak frequency further increases, the amplitude will get bigger as well. Secondly, when the superficial gas velocity increases to 10.0 m/s, only one obvious predominant It can be immediately found that visible distinctions exist with an increase of u SG . In Figure 7a, three predominant frequencies could be observed with the amplitude decreasing as the peak frequency increases. When the superficial gas velocity is 5.0 m/s, four predominant frequencies exist as shown in Figure 7b, which could prove that some differences may exist for the variations of the period of these two cases as mentioned above. The amplitudes of the minimum peak frequency and the maximal peak frequency are almost the same. While the amplitudes of second and third peak frequencies are relatively small. The condition of u SL = 7.0 m/s as depicted in Figure 7c seems to be similar to the condition of u SG = 5.0 m/s. However, the amplitude of the maximal peak frequency is larger than that of the minimum peak frequency. It was noted that the midpoint transverse displacement would no longer present the characteristic of periodicity when the superficial gas velocity is larger than 7.0 m/s. Some interesting phenomena will also occur for these conditions. Firstly, four predominant frequencies exist in the case of u SG = 8.0 m/s as shown in Figure 7d. The amplitudes of first and second peak frequencies are both very small. As the peak frequency further increases, the amplitude will get bigger as well. Secondly, when the superficial gas velocity increases to 10.0 m/s, only one obvious predominant frequency could be found as depicted in Figure 7f. It could be noted that the PSD for the case of u SG = 7.0 m/s and u SG = 9.0 m/s are similar to each other. Merely one main peak with small amplitude exists in the case of u SG = 7.0 m/s. Then it seems that conspicuous discrepancies exist in the vibration shapes for these two cases, which means that the analysis of PSD of vibration acceleration could really reflect the dynamic responses of the piping system.

The Dynamic Responses of E = 120 GPa
The bifurcation diagram of E = 120 GPa is shown in Figure 8. It can be found that the dynamical behavior of u SG = 8.0 m/s seems to be different from other conditions. Yet it is much similar to the situation of u SG = 10.0 m/s for E = 70 GPa. In the same way, we will discuss the dynamic responses of the piping system of six conditions as shown in Figure 9. frequency could be found as depicted in Figure 7f. It could be noted that the PSD for the case of uSG = 7.0 m/s and uSG = 9.0 m/s are similar to each other. Merely one main peak with small amplitude exists in the case of uSG = 7.0 m/s. Then it seems that conspicuous discrepancies exist in the vibration shapes for these two cases, which means that the analysis of PSD of vibration acceleration could really reflect the dynamic responses of the piping system.

The Dynamic Responses of E = 120 GPa
The bifurcation diagram of E = 120 GPa is shown in Figure 8. It can be found that the dynamical behavior of uSG = 8.0 m/s seems to be different from other conditions. Yet it is much similar to the situation of uSG = 10.0 m/s for E = 70 GPa. In the same way, we will discuss the dynamic responses of the piping system of six conditions as shown in Figure  9.  frequency could be found as depicted in Figure 7f. It could be noted that the PSD for the case of uSG = 7.0 m/s and uSG = 9.0 m/s are similar to each other. Merely one main peak with small amplitude exists in the case of uSG = 7.0 m/s. Then it seems that conspicuous discrepancies exist in the vibration shapes for these two cases, which means that the analysis of PSD of vibration acceleration could really reflect the dynamic responses of the piping system.

The Dynamic Responses of E = 120 GPa
The bifurcation diagram of E = 120 GPa is shown in Figure 8. It can be found that the dynamical behavior of uSG = 8.0 m/s seems to be different from other conditions. Yet it is much similar to the situation of uSG = 10.0 m/s for E = 70 GPa. In the same way, we will discuss the dynamic responses of the piping system of six conditions as shown in Figure  9.   Figure 9d. The amplitude will continue to increase as time goes on. In summary, like the case of E = 70 GPa, with the increase of superficial gas velocity, the dynamic responses will vary from periodic-like motion to a kind of motion whose amplitude increases with time. However, with the further increase of superficial gas velocity, the dynamic response will regain the periodic-like motion yet the amplitude will become larger.
The power spectrum densities (PSD) of the vibration acceleration for various superficial gas velocities E = 120 GPa and uSL = 1.0 m/s are shown in Figure 10.  Figure 9c, periodic vibration could also be observed which is similar to the variation of u SG = 7.0 m/s for E = 70 GPa as shown in Figure 6c yet with a shorter cycle. The dynamic response of u SG = 8.0 m/s is still distinctive as shown in Figure 9d. The amplitude will continue to increase as time goes on. In summary, like the case of E = 70 GPa, with the increase of superficial gas velocity, the dynamic responses will vary from periodic-like motion to a kind of motion whose amplitude increases with time. However, with the further increase of superficial gas velocity, the dynamic response will regain the periodic-like motion yet the amplitude will become larger.
The power spectrum densities (PSD) of the vibration acceleration for various superficial gas velocities E = 120 GPa and u SL = 1.0 m/s are shown in Figure 10.  Figure 9c, periodic vibration could also be observed which is similar to the variation of uSG = 7.0 m/s for E = 70 GPa as shown in Figure 6c yet with a shorter cycle. The dynamic response of uSG = 8.0 m/s is still distinctive as shown in Figure 9d. The amplitude will continue to increase as time goes on. In summary, like the case of E = 70 GPa, with the increase of superficial gas velocity, the dynamic responses will vary from periodic-like motion to a kind of motion whose amplitude increases with time. However, with the further increase of superficial gas velocity, the dynamic response will regain the periodic-like motion yet the amplitude will become larger.
The power spectrum densities (PSD) of the vibration acceleration for various superficial gas velocities E = 120 GPa and uSL = 1.0 m/s are shown in Figure 10. In the following discussion, "the first" peak frequency will be employed to represent the smallest peak frequency. "The second" will be the peak frequency which is just larger than "the first". The rest can be done in the same manner. Two predominant frequencies could be observed In Figure 10a Figure 11. It can be immediately found that the piping system may undergo a kind of chaotic-like motion for uSG = 9.0 m/s and E = 70 GPa in Figure 11a which is different from the dynamic behaviors of uSG = 10.0 m/s for E = 70 GPa and uSG = 8.0 m/s for E = 120 GPa. It should be noted that the chaoticlike motion does not have to occur for the higher gas flow rate as the slug parameters would affect the dynamic behaviors of the piping system a lot, which means that the slug parameters induced by the intermittent characteristics should be particularly considered to obtain the dynamic behaviors. In the following discussion, "the first" peak frequency will be employed to represent the smallest peak frequency. "The second" will be the peak frequency which is just larger than "the first". The rest can be done in the same manner. Two predominant frequencies could be observed In Figure 10a The phase portraits of the piping system are employed to explain this phenomenon as shown in Figure 11. It can be immediately found that the piping system may undergo a kind of chaotic-like motion for u SG = 9.0 m/s and E = 70 GPa in Figure 11a which is different from the dynamic behaviors of u SG = 10.0 m/s for E = 70 GPa and u SG = 8.0 m/s for E = 120 GPa. It should be noted that the chaotic-like motion does not have to occur for the higher gas flow rate as the slug parameters would affect the dynamic behaviors of the piping system a lot, which means that the slug parameters induced by the intermittent characteristics should be particularly considered to obtain the dynamic behaviors.

Conclusions
Based on slug flow parameters, the dynamic model for a simply supported piping system conveying two-phase slug flow is established to analyze the dynamic responses of the piping system under several pipe materials and superficial gas and liquid velocities. Some conclusions are listed as follows: (1) For various pipe materials, the dynamic responses of the piping system perform different patterns. The variations of the midpoint transverse displacement for E = 70 GPa and E = 120 GPa will vary from periodic-like motion to a kind of motion whose amplitude increases as time goes on with the increase of superficial gas velocity. However, for E = 120 GPa the dynamic response will regain the periodic-like motion with the further increase of superficial gas velocity.
(2) The dynamic responses have certain relations with the vibration acceleration. Firstly, if there are four predominant frequencies in the PSDs of the vibration acceleration, the piping system will perform periodic-like motion yet with the amplitude varying as time goes on. Secondly, if there are three predominant frequencies with the first peak frequency having the largest amplitude, the dynamic response will present the characteristic of periodic while the vibration amplitude varies with time in each periodic. Finally, the amplitude of the midpoint transverse displacement may continue to increase. For this case, if there is only one peak frequency that occupies the dominant status, the dynamic responses will follow certain rules. Otherwise, the piping system may perform a kind of chaotic-like motion.

Conclusions
Based on slug flow parameters, the dynamic model for a simply supported piping system conveying two-phase slug flow is established to analyze the dynamic responses of the piping system under several pipe materials and superficial gas and liquid velocities. Some conclusions are listed as follows: (1) For various pipe materials, the dynamic responses of the piping system perform different patterns. The variations of the midpoint transverse displacement for E = 70 GPa and E = 120 GPa will vary from periodic-like motion to a kind of motion whose amplitude increases as time goes on with the increase of superficial gas velocity. However, for E = 120 GPa the dynamic response will regain the periodic-like motion with the further increase of superficial gas velocity.
(2) The dynamic responses have certain relations with the vibration acceleration. Firstly, if there are four predominant frequencies in the PSDs of the vibration acceleration, the piping system will perform periodic-like motion yet with the amplitude varying as time goes on. Secondly, if there are three predominant frequencies with the first peak frequency having the largest amplitude, the dynamic response will present the characteristic of periodic while the vibration amplitude varies with time in each periodic. Finally, the amplitude of the midpoint transverse displacement may continue to increase. For this case, if there is only one peak frequency that occupies the dominant status, the dynamic responses will follow certain rules. Otherwise, the piping system may perform a kind of chaotic-like motion.

Acknowledgments:
The authors would like to thank the editors and reviewers for their valuable and constructive comments.

Conflicts of Interest:
The authors declare no conflict of interest.

Nomenclature
Ai sectional area of the inner pipe, m 2 E Young's modulus of the pipe, GPa I inertial moment of cross-section area, m 4 L length of the pipe, m m L (x,t) mass of liquid phase per unit length at the coordinate x and moment t, kg/m m G (x,t) mass of gas phase per unit length at the coordinate x and moment t, kg/m m P mass of fluid per unit length, kg/m Q G volume flow rates of the gas, m 3 /s Q L , volume flow rates of the liquid, m 3 /s u L (x,t) local velocity of liquid phase at the coordinate x and moment t, m/s u G (x,t) local velocity of gas phase at the coordinate x and moment t, m/s u SG superficial gas velocity, m/s u SL superficial liquid velocity, m/s y transverse displacement of the pipe, m Greek symbols ρ density, kg/m 3 Subscripts L liquid phase G gas phase